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<span class="kw">set </span><font color="Maroon" title="c3">x</font> =  <a href="heyting3.html#K3" title="HEYTING3:func.3">PFCrt</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font>;<br/>
<a NAME="E1:26_1_1"/>
 <a href="heyting3.html#K3" title="HEYTING3:func.3">PFCrt</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">k</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">t</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="heyting3.html#K3" title="HEYTING3:func.3">PFCrt</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font> or <font color="Maroon" title="c4">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">k</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:26_1_1_1"/>
<font color="Maroon" title="c4">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="heyting3.html#K3" title="HEYTING3:func.3">PFCrt</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">k</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">m</font> being   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:26_1_1_1"/><span class="lab"><font color="Green" title="E12">A1</font></span>: 
<font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">and </span><br/><a NAME="E4:26_1_1_1"/><span class="lab"><font color="Green" title="E13">A2</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c2">k</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">t</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="heyting3.html#D3" target="_self" title="HEYTING3:def.3">Def3</a></span>;<br/>
<a NAME="E5:26_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a></span>;<br/>
<a NAME="E6:26_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26_1_1_1"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:26_1_1_1"/>
<font color="Maroon" title="c4">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">k</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_1_1_1"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="ref" href="zfmisc_1.html#T129" title="ZFMISC_1:th.129">ZFMISC_1:129</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:26_1_1"/>
 <a href="heyting3.html#K3" title="HEYTING3:func.3">PFCrt</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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